Lecture Notes

Drawbacks of Riemann integral: The set of Riemann integrable functions turns out to be a rather unsatisfactorily small class of functions (textbook Tao).

Exceptions of Lebesgue integral: “If one assumes the axiom of choice, then there are still some pathological functions one can construct which cannot be integrated by the Lebesgue integral, but these functions will not come up in real-life applications.” (textbook Tao).

Question: what are the pathological functions and is it possible that someday we find some pathological functions that do have real-life applications?

Question: why “outer” measure instead of “inner” measure?

Ans: they are dual approaches

Question: why do we defined measure based on sigma-algebra?

Question: why do we require the covering to be at most countable in the definition of outer measure?

Question: any box has $m^{*}(\cdot)=\text{volume}$?

Ans: $m^{*}(B)=vol(\bar{B})=vol(B^{\circ})=\text{volume}$

Definition(caratheory criterion, recap): a set $E$ is measurable if $\forall A\subset\mathbb{R}^n$

$$ m^{*}(A)=m^{*}(A\cap E)+m^{*}(A\backslash E) $$

Definition(new criterion in homework 2): a set $E$ is measurable if $\forall\epsilon>0,\exists U\subset\mathbb{R}^n$ such that 1) $U$ is an open set; 2) $E\subset U$; 3) $m^{*}(U\backslash E)<\epsilon$

Proposition(for the discussion topic in lecture 4): if a bounded set $E$ is measurable under the caratheory criterion, then $E$ is also measurable under the new criterion in homework 2.

From my point of view, outer measure is similar to open cover for compact sets. I want to develop a different type of measure on my own. I just don't like the supremum/infimum thing. They are not elegant symbols anyway. I try not to review too much previous literature but enjoy developing a new measure on my own.

(Reference: Tao, Measure & Analysis II)

• What does it mean for a subset of $\mathbb{R}^d$ to be measurable?
• If a set is measurable, how to define its measure?
• What nice properties does measure obey?

It turns out that there are some “ill” subsets of $\mathbb{R}^d$ that are always unmeasurable.

Indicator function: Given a set $T$ and its subset $S\subset T$, we can define an indicator function $1_S:T\rightarrow\{0,1\}$ such that if $x\in S,1_S(x)=1$; if $x \overline{\in} S,1_S(x)=0$.

We can choose $x_{\lambda}$, a family of elements, from a family of nonempty sets $(X_{\lambda})_{\lambda\in\Lambda}$.

Measure concerns infinity. So we must define the algebra concerning infinity 👇

Infinity rules:

• $(+\infty)+x=x+(+\infty)=(+\infty),\forall x\in[0,+\infty]$
• $(+\infty)\cdot x=x\cdot(+\infty)=(+\infty),\forall x\in(0,+\infty]$
• $(+\infty)\cdot 0=0\cdot(+\infty)=0$
• $x<(+\infty),\forall x\in[0,+\infty)$

These rules guarantee that multiplication functions are “upward continuous”. For example, we define $m_{0}(x)=0\cdot x$, then $\forall x\in[0,+\infty],m_{0}(x)=0$ (continuous). If we use the rule “$(+\infty)\cdot 0=0\cdot(+\infty)=1$” then $m_{0}(x)$ is no longer continuous.

We are also interested in other operations concerning the notion of infinity — the infinite sums (the simpler version of integrals). We find that some divergent sums become convergent if we introduce infinity to the real line. Do all sums in the form of $\sum_{n=1}^{\infty}x_n$ converge in this extended setting?

$$ \sum_{n=1}^{\infty}x_n:=\sup_{F\subset\mathbb{N},F\text{ finite}}\sum_{n\in F}x_n $$

Note that this definition only works on $[0,+\infty]$. We take supremum since all $x_n\geq 0$.

Statement: Given $(x_{n,m})_{n,m\in\mathbb{N}}$ where $\forall x_{n,m}\in[0,+\infty]$, then

$$ \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{n,m}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}x_{n,m} $$

Remark: $\sum_{(n,m)\in\mathbb{N}^2}x_{n,m}$ represents an arbitrary summation method over $\mathbb{N}^2$. This is compatible with our new definition of infinite sums 👆.

We can also formalize this representation with a bijective function $\phi:\mathbb{N}^2\rightarrow\mathbb{N}^2$. Rewrite the summation as $\sum_{(n,m)\in\mathbb{N}^2,\phi}x_{n,m}:=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{\phi(n,m)}$. But this notation does not provide better ways to prove the statement.

Alternative statement: Given $(x_{n,m})_{n,m\in\mathbb{N}}$ where $\forall x_{n,m}\in[0,+\infty]$, then any method to sum up $x_{n,m}$ ends up with the same value.

Proof:

Failed attempt: $\sum_{n=1}^{\infty}\sum_{m=1}^{M}x_{n,m}\stackrel{?}{=}\sum_{m=1}^{M}\sum_{n=1}^{\infty}x_{n,m}$. It would be hard to extend from this “partial finite” equality to a “double infinite” one. (This attempt provides a hint for the second part of the following sketch.)

Sketch:

(1) $\sum_{(n,m)\in F\subset \{1,\dots,N\}\times \{1,\dots,N\}}x_{n,m}\leq \sum_{n=1}^{N}\sum_{m=1}^{N}x_{n,m}\leq \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{n,m}\Longrightarrow$ by definition of infinite sum we have $\sum_{(n,m)\in\mathbb{N}^2}x_{n,m}= \sup\dots\leq \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{n,m}$.

(2) (refer to the failed attempt, I call this trick “double anchors”) $\sum_{n=1}^{N}\sum_{m=1}^{\infty}x_{n,m}\leq \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}\Longrightarrow \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{n,m}\leq \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}$. We can just show that $\sum_{n=1}^{N}\sum_{m=1}^{\infty}x_{n,m}\leq \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}$. Fix $N$, relax the second summation, and get a finite sum $\sum_{n=1}^{N}\sum_{m=1}^{M}x_{n,m}\leq \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}$. Now apply order limit theorem twice we derive $\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}x_{n,m}\leq \sum_{(n,m)\in\mathbb{N}^2}x_{n,m}$.

Statement I: let $A,B$ be sets (possibly infinite or uncountable) and $(x_{n,m})_{n\in A, m\in B}\in[0,+\infty]$, then

$$ \sum_{(n,m)\in A\times B}=\sum_{n\in A}\sum_{m\in B}x_{n,m}=\sum_{m\in B}\sum_{n\in A}x_{n,m} $$

Statement II: let $A_1,A_2,\dots,A_k$ be sets (possibly infinite or uncountable) and $(x_{n_1,n_2,\dots,n_k})_{n_1\in A_1,\dots,n_k\in A_k}\in[0,+\infty]$, then

$$ \sum_{(n_1,\dots,n_k)\in A_1\times\dots\times A_k}x_{n_1,n_2,\dots,n_k}=\sum_{n_1\in A_1}\sum_{n_2\in A_2}\dots\sum_{n_k\in A_k}x_{n_1,n_2,\dots,n_k}=\dots $$

In this section I begin to establish the basic ideas of measure by equivalent classes. Here I also assume preliminary knowledge in point set topology.

(Reference: Tao, Measure & Analysis II)

length of an interval $I=[a,b],[a,b),(a,b],(a,b)\Longrightarrow |I|=|b-a|$

any box $B=I_1\times\dots\times I_d$ in $\mathbb{R}^d$ has measure (generalized volume) $|B|=|I_1|\times\dots\times|I_d|$

In this section, I introduce something new. Make use of my previous note on summation.

Lemma: in $\mathbb{R}$, any open set is a union of open intervals

proof sketch: given any open set $O\subset\mathbb{R}$

(1) define an equivalence relation over $\mathbb{R}$. $\forall x,y\in O,x\sim y$ if $\exists(a,b)\subset O$ such that $x,y\in(a,b)$

(2) now any point in $O$ is in an equivalence class, prove that any equivalent class $E\in O/\sim$ is an open interval, qed

Definition: $A\subset\mathbb{R}^n$ is an open set if and only if $\forall a\in A,\exists r\in\mathbb{R}_{>0}$ such that open ball $B_r(a)\subset A$

Lemma: in $\mathbb{R}^n$, any open set is a countable union of open boxes

proof sketch:

(1) union: just take boxes inside the promised open ball around any point and then we can prove this follow what we did in $\mathbb{R}^n$

(2) countable: all rational balls (rational center & rational radius) form a topological base of $\mathbb{R}^n$

Given any open set $A\subset\mathbb{R}^n$, define an equivalence relation $\forall x,y\in A,x\sim y$ if $\exists$ open box $B$ such that $x,y\in B$. Then with the equivalence relation we derive a partition of $A$, i.e. the collection of all equivalence classes: $A/\sim$.

Define the measure of $A$ as $m(A):=\sum_{[x]\in A/\sim} m|[x]|$

Properties:

• Translation invariance (since the equivalent class, i.e. open boxes are translation invariant)
• Monotonicity: $E\subset F\Longrightarrow m(E)\leq m(F)$

When $n=1$, our target is just the real line, $\mathbb{R}$.

Lemma: in $\mathbb{R}$, the equivalence classes defined above are themselves open intervals

However, it is hard to generalize this lemma to higher dimensions. The equivalence classes in arbitrary $\mathbb{R}^n$ may not be higher-dimension open boxes.

Given any closed set $K\subset\mathbb{R}$, define an equivalence relation $\forall x,y\in K,x\sim y$ if $\exists B=I_1\times\dots\times I_d\subset K$ such that $x,y\in B$

This measure doesn't work well in higher dimension if we do not change the definition of equivalent relation. It turns out that the supremum/infimum thing is essential because we cannot expect measurable sets in higher dimensions come in a “good shape”. Previous equivalent classes (open boxes) does not fit them well. Supremum/infimum helps us to approximate the sets.

The Lebesgue outer measure of a set $A\subset\mathbb{R}$ is

$$ m^{*}A=\inf\left\{\sum_{k}|I_k|:\{I_k\}\text{ is a covering of }A\text{ by open intervals}\right\} $$

A box $B$ is the cartesian product of $n$ intervals: $B=I_1\times\dots\times I_n$. The Lebesgue outer measure of a set $A\subset\mathbb{R}^n$ is

$$ m^{*}A=\inf\left\{\sum_{k}\left|B_k\right|:\{B_k\}\text{ covers }A\right\} $$

Properties

• $m^{*}\emptyset=0$ because by definition any small covering covers emptyset so the infimum is 0
• (positivity) $0\leq m^{*}(\Omega)\leq+\infty$ for every measurable set $\Omega$
• (monotonicity) $A\subset B\subset\mathbb{R}\Longrightarrow m^{*}A\leq m^{*}B$ because every covering of $B$ must also cover $A$
• (subadditivity) $A=\cup_{n=1}^{\infty}A_n\Longrightarrow m^{*}A\leq\sum_{n=1}^{\infty}m^{*}A_n$
• (translation invariance) $\Omega\subset\mathbb{R}^n,x\in\mathbb{R}^n\Longrightarrow m^{*}(x+\Omega)=m^{*}(\Omega)$

Proof of subadditivity

Lemma: $\inf\{\sum_{i=1}^{N}a_i:a_i\in A_i\}=\sum_{i=1}^{N}\inf A_i$ ($A_i\subset\mathbb{R}$ is bounded below)

$\because$ pick any covering $\{I_k\}_n$ of $A_n$'s, we form a new covering $\{\{I_k\}_n\}$ of $A$

$\therefore\sum_{n,k}|I_{k,n}|\geq m^{*}A$

$\therefore \sum_{n}m^{*}A_n\geq m^{*}A$

Certain sets badly behaved with respect to outer measure. We must exclude the pathological sets with the concept of measurability. Recall 👇

• Borel property (open/close)
• Complementarity
• Boolean/$\sigma$ algebra property (union/intersection)
• Translation invariance
• Compatible with our definition of length, area and volume

Definition (ver 1): The set $E\subset\mathbb{R}^n$ is (Lebesgue) measurable iff. for any set $X\subset\mathbb{R}^n$

$$ m^{*}X=m^{*}(X\cap E)+m^{*}(X\cap E^c) $$

Definition (ver 2): The set $E\subset\mathbb{R}^n$ is (Lebesgue) measurable iff. $\forall\epsilon>0$, there exists an open set $U$ such that

$$ E\subset U\text{ and }m^{*}(U/E)<\epsilon $$

Proposition: the two measurability definitions are equivalent

Measurable sets are scalpels and bandages.

Summary of tricks:

1. Use finite operations to approach infinite ones
2. Use $\sup,\inf$ to bypass limit/infinite sums/other infinite operations (see the countable additivity example, Tao analysis II lemma 7.4.8)
3. $\epsilon/2^n$ trick

For the complete note, please refer to my Google drive.

Homework folder: https://drive.google.com/drive/folders/1TjC_s140VMZ1tHT4UanLMLWzYbZi5E7Z?usp=sharing

Homework 1

https://drive.google.com/file/d/1WOPDRyQWb4DAcxxjcN1Mwh8nwfqNLrTZ/view?usp=sharing

Homework 2

https://drive.google.com/file/d/1kgrGQJLadXnqPXJB49yGb2NHtyXjZJyq/view?usp=sharing

Homework 3

https://drive.google.com/file/d/1tNUhlKKdrHXal4Fi-YqoI_wv6594LWnf/view?usp=sharing

Homework 4

Comparison between Riemann integral and Lebesgue integral

Let's first review several definitions in Riemann integral.

Definition: A partition $P$ of $[a,b]$ is a finite set of points from $[a,b]$ that includes $a$ and $b$. We can just list the points of partition $P=\{x_0,x_1,x_2,\dots,x_n\}$

$$ a=x_0<x_1<x_2<\cdots<x_n=b $$

Let $m_k=\inf\{f(x):x\in[x_{k-1},x_k]\}$ and $M_k=\sup\{f(x):x\in[x_{k-1},x_k]\}$. The lower sum and the upper sum of $f$ with respect to $P$ is given by $$ L(f,P)\sum_{k=1}^{n}m_k\cdot(x_k-x_{k-1}),\quad U(f,P)\sum_{k=1}^{n}M_k\cdot(x_k-x_{k-1}) $$

Definition: Let $\mathcal{P}$ be the collection of all possible partitions of the interval $[a,b]$. The upper integral and lower integral of $f$ is defined to be $$U(f)=\inf\{U(f,P):P\in\mathcal{P}\},\quad L(f)=\sup\{L(f,P):P\in\mathcal{P}\}$$

Definition(Riemann Integrability): A bounded function $f$ defined on the interval $[a,b]$ is Riemann-integrable if $U(f)=L(f)$. In this case, we define $\int_{a}^{b}f$ or $\int_{a}^{b}f(x)dx$ to be this common value; namely,

$$\int_a^bf=U(f)=L(f)$$

Theorem: Let $f$ be a bounded function defined on the interval $[a,b]$. Then, $f$ is Riemann-integrable if and only if the set of points where $f$ is not continuous has measure zero.

As Prof. Zhou mentioned, using simple functions to introduce Lebesgue integral is a traditional approach. Let's pick a simple function to see the difference between Riemann integral and Lebesgue integral.

Example: Consider the Dirichlet function:

$$ \mathbf{1}_{\mathbb{Q}}(x)= \begin{cases} 1\quad&x\in\mathbb{Q} \\0 \quad&x\in\mathbb{R}\backslash\mathbb{Q} \end{cases} $$

Integrate this function on $[0,1]$. Since the function is discontinuous everywhere. By the above theorem we know we cannot integrate the function with Riemann integral. Dirichlet function is a simple function. So we can integrate it with Lebesgue integral.

$$ \int_{\mathbb{Q}}\mathbf{1}_{\mathbb{Q}}(x)dx =1\times\text{m}(\mathbb{Q}[0,1])=0 $$

Reference: Stephen Abbott. (2010). Understanding Analysis.

Homework 4

https://drive.google.com/file/d/13LHe9rVFRnsDRmUUFgccFJ9D_NQwik8u/view?usp=sharing

Homework 5

https://drive.google.com/file/d/1XKaPiGecXvbfXuEt8nMrLe-mzR3xff02/view?usp=sharing

Homework 6

https://drive.google.com/file/d/1sVQzddDbmOyto7RX41eDsbjTOydE0n5Q/view?usp=sharing

Homework 7

https://drive.google.com/file/d/1RJSeEpIKoGRP5qVWHsAO0InZK9WL-wey/view?usp=sharing

Homework 8

Comments about Littlewood's three principles

There are four equivalent viewpoints of random surfaces: Brownian sphere, Peanosphere, Liouville quantum gravity sphere, Conformal field theory (it is highly non-trivial to show their equivalence).

First we introduce Brownian motion (Wiener process).

Definition[Approximate Wiener process]: Let $\xi_1,\xi_2,\dots$ be a sequence of iid random variables with mean 0 and variance 1. For every $n\in\{1,2,3,\dots\}$, define a continuous-time stochastic process $\{W_n(t)\}_{t\geq0}$ by $W_n(t)=\frac{1}{\sqrt{n}}\sum_{1\leq j\leq \lfloor nt\rfloor}\xi_j$.

Property: $n\rightarrow\infty,(W_n(t)-W_n(s))\approx N(0,t-s)$.

Definition[Wiener process]: $W(t):=\lim\limits_{n\rightarrow\infty}W_n(t)$.

Definition[Brownian bridge]: A brownian bridge is a continuous stochastic process $B_t$ defined in the interval $[0,T]$ by a Wiener process $W_t$, such that $B_t:=(W_t|W_T=W_0=0)$.

We can easily define a brownian bridge by $W(t)-\frac{t}{T}W(T),t\in[0,T]$.

Definition[Brownian excursion]: A brownian excursion is a brownian bridge conditioned to be positive.

Definition[Topological path]: Fix a topology space $(X,\theta_X)$. A topological path in $X$ is a continuous mapping $\gamma:[a,b]\subset\mathbb{R}\rightarrow X$.

Definition[Real/continuum tree]: a compact metric space $(X,d)$ is a real tree if (1) $\forall x,y\in X$, there exists a shortest path $ [ [ x , y ] ] $ from $x$ to $y$ with length $d(x,y)$ ; (2) $\forall x,y\in X$, the only non-self-intersecting path from $x$ to $y$ is $ [ [ x , y ] ] $.

Example: we can construct a real tree with an undergraph of a function. Please refer to Christina Goldschmidt's lecture https://www.stats.ox.ac.uk/~goldschm/WarwickLecture2.pdf

How can we generate a random tree? There are several methods. For example, we can generate the tree via a branching process (Galton–Watson process).

1. What is a random surface? Scott Sheffield. Retrieved from https://arxiv.org/pdf/2203.02470.pdf

2. Brownian Motion. Steven Lalley. Retrieved from https://galton.uchicago.edu/~lalley/Courses/313/BrownianMotionCurrent.pdf

3. Lecture 2: The continuum random tree (continued). Christina Goldschmidt. Retrieved from https://www.stats.ox.ac.uk/~goldschm/WarwickLecture2.pdf

4. The Continuum Random Tree I. David Aldous. Retrieved from https://projecteuclid.org/journals/annals-of-probability/volume-19/issue-1/The-Continuum-Random-Tree-I/10.1214/aop/1176990534.full

5. The Continuum Random Tree II: an overview. David Aldous. Retrieved from https://www.stat.berkeley.edu/~aldous/Papers/me55.pdf

6. The Continuum Random Tree III. David Aldous. Retrieved from https://projecteuclid.org/journals/annals-of-probability/volume-21/issue-1/The-Continuum-Random-Tree-III/10.1214/aop/1176989404.full

7. Gwynne, E., Miller, J.P., Sheffield, S. The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity. https://arxiv.org/pdf/1705.11161.pdf

8. Gwynne, Ewain, Jason Miller and Scott Sheffield. “Harmonic functions on mated-CRT maps.” Electronic journal of probability, 24, 58 (May 2019) https://dspace.mit.edu/bitstream/handle/1721.1/126714/1807.07511.pdf?sequence=2&isAllowed=y

9. Jason Miller - 1/4 Equivalence of Liouville quantum gravity and the Brownian map. Institut des Hautes Études Scientifiques (IHÉS). https://www.youtube.com/watch?v=NB9iZ8ZX4dQ

http://assets.press.princeton.edu/chapters/s8008.pdf